The Born-Oppenheimer Approximation

• The Hamiltonian for several nuclei and electrons would be

  

  where A runs over nuclei, i over electrons. Note that unlike an atom in free space, we now have a term in nuclear motion. Wave function would be

  .

• Nuclei are much heavier than electrons (at least 2000 times). Decouple problem: assert that there is no coupling between electornic motion (fast) and nuclear motion (slow).

• Write wave function as , where we have a different electronic wave function for each value of the parameters RA - coordinates of fixed nuclei. Eigenfunction of

  

• Electronic Schrödinger equation

  

  the function E(RA) defines the potential energy surface for nuclear motion.

• Get rotational and vibrational energy levels for the nuclear part of the problem, by solving

  

  where is the nuclear motion wave function. Also determines chemical dynamics.

• Note that this is an approximation - properly speaking there are just eigenvalues of the full Hamiltonian, representing "rovibronic" levels. But a very good one.

• Fixing the nuclei has a drastic effect on the symmetry of the system!

Hartree-Fock Theory

• Even with the Born-Oppenheimer approimxation, we can make no progress on analytical solution of the Schrödinger equation.

• Start simple: assume traditional molecular orbital ideas hold. For a system of 2N electrons, assume tha the wave function is given by allocating electrons in pairs to molecular orbitals .

• Physically, as we shall see, this is like assuming that the electrons interact only with the average potential of all the other electrons. Independent-particle model, or mean-field theory.

• Assume that the MOs are orthonormal (and here real)

  

• Trial wave function is a Slater determinant

  

  Substitute into the variation principle, and make stationary with respect to the form of the MOs.

• Need to evaluate

  

  for this form of . Helped here by the orthonormality of the MOs and by the form of H.

• H contains zero-, one-, and two-electron operators only. In the integral over all space, we have contributions from the coordinates of N-2 electrons that are just products of integrals

  

  so thanks to orthnormality we end up with only one- and two-electron integrals:

  

  where

  

• The expression

  

  is the energy for a given guess at the MOs. We optimize the MOs by minimizing this energy (variational method).

• Minimizing the energy requires satisfying the Hartree-Fock equations

  

  is the Fock operator.

• Notes the sum over all k here. The solutions depend on themselves! This is not a simple eigenvalue problem.

• Self-consistent field (SCF) approach: guess a sert of MOs, construct F, diagonalize, and use its eigenvectors as a (hopefully) better guess at the MOs.

Solving the SCF equations

• For atoms, the symmetry of the system factors the SCF equations into an angular part (spherical harmonics again!) and a radial part. The radial equations can be solved numerically (one-dimensional problem).

• For molecules, the lower symmetry does not permit any factorization in general. Numerical methods have been used for diatomic molecules, but becomes expensive and cumbersome for polyatomic systems.

• How do we then express the MOs? Expand them in a fixed basis set, just like the traditional LCAO (linear combination of atomic orbitals) reasoning used in qualitative MO theory.

The SCF equations

• Let us expand the unknown MOs linearly in a set of basis functions:

  

  Here, can be any fucntions deemed appropriate. They are often approximations to atomic orbitals on the individual centres, and are often termed "atomic orbitals" (AOs).

• Note that the AOs are not required to be orthonormal:

  

• The SCF equations now become

  

  where,

  

SCF Calculations

• Choose a basis .

• Calculate the integrals over this basis (store on disk).

• Guess the MO coefficients C.

• Construct D, then F. Solve the equations for a new C. If it does not agree with the previous estimate, iterate until it does.